MinT - Best Known (157−73, 157, s)-Nets in Base 3

Best Known (157−73, 157, s)-Nets in Base 3

(157−73, 157, 69)-Net over F₃ — Constructive and digital

Digital (84, 157, 69)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 57, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]
2. digital (27, 100, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(157−73, 157, 100)-Net over F₃ — Digital

Digital (84, 157, 100)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−73, 157, 799)-Net in Base 3 — Upper bound on s

There is no (84, 157, 800)-net in base 3, because

1 times m-reduction [i] would yield (84, 156, 800)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 278 669226 454703 288485 736840 013888 016960 712576 922609 758210 228764 990777 978625 > 3¹⁵⁶ [i]